Hilbert-Schmidt Operators vs. Integrable Systems of Elliptic Calogero-Moser Type III. The Heun Case
The Heun equation can be rewritten as an eigenvalue equation for an ordinary differential operator of the form −d²/dx²+V(g;x), where the potential is an elliptic function depending on a coupling vector g ∈ R⁴. Alternatively, this operator arises from the BC1 specialization of the BCN elliptic nonrel...
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| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Date: | 2009 |
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Інститут математики НАН України
2009
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/149153 |
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| Cite this: | Hilbert-Schmidt Operators vs. Integrable Systems of Elliptic Calogero-Moser Type III. The Heun Case / Simon N.M. Ruijsenaars // Symmetry, Integrability and Geometry: Methods and Applications. — 2009. — Т. 5. — Бібліогр.: 20 назв. — англ. |
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Ruijsenaars, Simon N.M. 2019-02-19T17:47:23Z 2019-02-19T17:47:23Z 2009 Hilbert-Schmidt Operators vs. Integrable Systems of Elliptic Calogero-Moser Type III. The Heun Case / Simon N.M. Ruijsenaars // Symmetry, Integrability and Geometry: Methods and Applications. — 2009. — Т. 5. — Бібліогр.: 20 назв. — англ. 1815-0659 2000 Mathematics Subject Classification: 33E05; 33E10; 46N50; 81Q05; 81Q10 https://nasplib.isofts.kiev.ua/handle/123456789/149153 The Heun equation can be rewritten as an eigenvalue equation for an ordinary differential operator of the form −d²/dx²+V(g;x), where the potential is an elliptic function depending on a coupling vector g ∈ R⁴. Alternatively, this operator arises from the BC1 specialization of the BCN elliptic nonrelativistic Calogero-Moser system (a.k.a. the Inozemtsev system). Under suitable restrictions on the elliptic periods and on g, we associate to this operator a self-adjoint operator H(g) on the Hilbert space H = L²([0,ω₁],dx), where 2ω₁ is the real period of V(g;x). For this association and a further analysis of H(g), a certain Hilbert-Schmidt operator I(g) on H plays a critical role. In particular, using the intimate relation of H(g) and I(g), we obtain a remarkable spectral invariance: In terms of a coupling vector c ∈ R⁴ that depends linearly on g, the spectrum of H(g(c)) is invariant under arbitrary permutations σ(c), σ ∈ S₄. This paper is a contribution to the Proceedings of the Workshop “Elliptic Integrable Systems, Isomonodromy Problems, and Hypergeometric Functions” (July 21–25, 2008, MPIM, Bonn, Germany). We would like to thank F. Nijhof f, B. Sleeman and K. Takemura for illuminating discussions and for supplying information about related literature. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Hilbert-Schmidt Operators vs. Integrable Systems of Elliptic Calogero-Moser Type III. The Heun Case Article published earlier |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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| title |
Hilbert-Schmidt Operators vs. Integrable Systems of Elliptic Calogero-Moser Type III. The Heun Case |
| spellingShingle |
Hilbert-Schmidt Operators vs. Integrable Systems of Elliptic Calogero-Moser Type III. The Heun Case Ruijsenaars, Simon N.M. |
| title_short |
Hilbert-Schmidt Operators vs. Integrable Systems of Elliptic Calogero-Moser Type III. The Heun Case |
| title_full |
Hilbert-Schmidt Operators vs. Integrable Systems of Elliptic Calogero-Moser Type III. The Heun Case |
| title_fullStr |
Hilbert-Schmidt Operators vs. Integrable Systems of Elliptic Calogero-Moser Type III. The Heun Case |
| title_full_unstemmed |
Hilbert-Schmidt Operators vs. Integrable Systems of Elliptic Calogero-Moser Type III. The Heun Case |
| title_sort |
hilbert-schmidt operators vs. integrable systems of elliptic calogero-moser type iii. the heun case |
| author |
Ruijsenaars, Simon N.M. |
| author_facet |
Ruijsenaars, Simon N.M. |
| publishDate |
2009 |
| language |
English |
| container_title |
Symmetry, Integrability and Geometry: Methods and Applications |
| publisher |
Інститут математики НАН України |
| format |
Article |
| description |
The Heun equation can be rewritten as an eigenvalue equation for an ordinary differential operator of the form −d²/dx²+V(g;x), where the potential is an elliptic function depending on a coupling vector g ∈ R⁴. Alternatively, this operator arises from the BC1 specialization of the BCN elliptic nonrelativistic Calogero-Moser system (a.k.a. the Inozemtsev system). Under suitable restrictions on the elliptic periods and on g, we associate to this operator a self-adjoint operator H(g) on the Hilbert space H = L²([0,ω₁],dx), where 2ω₁ is the real period of V(g;x). For this association and a further analysis of H(g), a certain Hilbert-Schmidt operator I(g) on H plays a critical role. In particular, using the intimate relation of H(g) and I(g), we obtain a remarkable spectral invariance: In terms of a coupling vector c ∈ R⁴ that depends linearly on g, the spectrum of H(g(c)) is invariant under arbitrary permutations σ(c), σ ∈ S₄.
|
| issn |
1815-0659 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/149153 |
| citation_txt |
Hilbert-Schmidt Operators vs. Integrable Systems of Elliptic Calogero-Moser Type III. The Heun Case / Simon N.M. Ruijsenaars // Symmetry, Integrability and Geometry: Methods and Applications. — 2009. — Т. 5. — Бібліогр.: 20 назв. — англ. |
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2025-12-07T17:17:20Z |
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2025-12-07T17:17:20Z |
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1850870690021900288 |